#554015

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(5,6,8,7)(10,13)(14,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31);(2,4)(10,13)(11,15)(17,19,22,27)(18,23,25,20)(21,29,30,26)(24,28,32,31);(1,2,3,4)(5,7,8,6)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32);(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(17,21,22,30)(18,24,25,32)(19,26,27,29)(20,28,23,31)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 461211);